+----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| v = | 0.3 | 0.5 | 0.7 | 1 | 1½ | 2 | 3 | 5 | 7 | 10 | 15 |
| | | | | | | | | | | | |
| [zeta] = |0.01215|0.01025|0.00944|0.00883|0.00836|0.00812|0.90788|0.00769|0.00761|0.00755|0.00750|
+----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
In using this value of [zeta] when v is not known, it is best to
proceed by approximation.
§ 98. _Darcy and Bazin's Expression for the Coefficient of
Friction._--Darcy and Bazin's researches have shown that [zeta] varies
very greatly for different degrees of roughness of the channel bed,
and that it also varies with the dimensions of the channel. They give
for [zeta] an empirical expression (similar to that for pipes) of the
form
[zeta] = a(1 + [beta]/m); (6)
where m is the hydraulic mean depth. For different kinds of channels
they give the following values of the coefficient of friction:--
+-------------------------------------------------+--------+------+
| Kind of Channel. | [alpha]|[beta]|
+-------------------------------------------------+--------+------+
| I. Very smooth channels, sides of smooth | | |
| cement or planed timber | .00294 | 0.10 |
| II. Smooth channels, sides of ashlar, brickwork,| | |
| planks | .00373 | 0.23 |
|III. Rough channels, sides of rubble masonry or | | |
| pitched with stone | .00471 | 0.82 |
| IV. Very rough canals in earth | .00549 | 4.10 |
| V. Torrential streams encumbered with detritus | .00785 | 5.74 |
+-------------------------------------------------+--------+------+
The last values (Class V.) are not Darcy and Bazin's, but are taken
from experiments by Ganguillet and Kutter on Swiss streams.
The following table very much facilitates the calculation of the mean
velocity and discharge of channels, when Darcy and Bazin's value of
the coefficient of friction is used. Taking the general formula for
the mean velocity already given in equation (2a) above,
v = c [root](mi),
where c = [root](2g/[zeta]), the following table gives values of c for
channels of different degrees of roughness, and for such values of the
hydraulic mean depths as are likely to occur in practical
calculations:--
Values of c in v = c[root](mi), deduced from Darcy and Bazin's Values.
+----------+-----------+----------+---------+----------+--------------+
| |Very Smooth| Smooth | Rough |Very Rough| Excessively |
| Mean | Channels, | Channels,|Channels,| Channels,|Rough Channels|
|Depth = m.| Cement. |Ashlar or | Rubble |Canals in | encumbered |
| | |Brickwork.| Masonry.| Earth. |with Detritus.|
+----------+-----------+----------+---------+----------+--------------+
| .25 | 125 | 95 | 57 | 26 | 18.5 |
| .5 | 135 | 110 | 72 | 36 | 25.6 |
| .75 | 139 | 116 | 81 | 42 | 30.8 |
| 1.0 | 141 | 119 | 87 | 48 | 34.9 |
| 1.5 | 143 | 122 | 94 | 56 | 41.2 |
| 2.0 | 144 | 124 | 98 | 62 | 46.0 |
| 2.5 | 145 | 126 | 101 | 67 | .. |
| 3.0 | 145 | 126 | 104 | 70 | 53 |
| 3.5 | 146 | 127 | 105 | 73 | .. |
| 4.0 | 146 | 128 | 106 | 76 | 58 |
| 4.5 | 146 | 128 | 107 | 78 | .. |
| 5.0 | 146 | 128 | 108 | 80 | 62 |
| 5.5 | 146 | 129 | 109 | 82 | .. |
| 6.0 | 147 | 129 | 110 | 84 | 65 |
| 6.5 | 147 | 129 | 110 | 85 | .. |
| 7.0 | 147 | 129 | 110 | 86 | 67 |
| 7.5 | 147 | 129 | 111 | 87 | .. |
| 8.0 | 147 | 130 | 111 | 88 | 69 |
| 8.5 | 147 | 130 | 112 | 89 | .. |
| 9.0 | 147 | 130 | 112 | 90 | 71 |
| 9.5 | 147 | 130 | 112 | 90 | .. |
| 10.0 | 147 | 130 | 112 | 91 | 72 |
| 11 | 147 | 130 | 113 | 92 | .. |
| 12 | 147 | 130 | 113 | 93 | 74 |
| 13 | 147 | 130 | 113 | 94 | .. |
| 14 | 147 | 130 | 113 | 95 | .. |
| 15 | 147 | 130 | 114 | 96 | 77 |
| 16 | 147 | 130 | 114 | 97 | .. |
| 17 | 147 | 130 | 114 | 97 | .. |
| 18 | 147 | 130 | 114 | 98 | .. |
| 20 | 147 | 131 | 114 | 98 | 80 |
| 25 | 148 | 131 | 115 | 100 | .. |
| 30 | 148 | 131 | 115 | 102 | 83 |
| 40 | 148 | 131 | 116 | 103 | 85 |
| 50 | 148 | 131 | 116 | 104 | 86 |
| [oo] | 148 | 131 | 117 | 108 | 91 |
+----------+-----------+----------+---------+----------+--------------+
§ 99. _Ganguillet and Kutter's Modified Darcy Formula._--Starting from
the general expression v = c[root]mi, Ganguillet and Kutter examined
the variations of c for a wider variety of cases than those discussed
by Darcy and Bazin. Darcy and Bazin's experiments were confined to
channels of moderate section, and to a limited variation of slope.
Ganguillet and Kutter brought into the discussion two very distinct
and important additional series of results. The gaugings of the
Mississippi by A. A. Humphreys and H. L. Abbot afford data of
discharge for the case of a stream of exceptionally large section and
or very low slope. On the other hand, their own measurements of the
flow in the regulated channels of some Swiss torrents gave data for
cases in which the inclination and roughness of the channels were
exceptionally great. Darcy and Bazin's experiments alone were
conclusive as to the dependence of the coefficient c on the dimensions
of the channel and on its roughness of surface. Plotting values of c
for channels of different inclination appeared to indicate that it
also depended on the slope of the stream. Taking the Mississippi data
only, they found
c = 256 for an inclination of 0.0034 per thousand,
= 154 " " 0.02 "
so that for very low inclinations no constant value of c independent
of the slope would furnish good values of the discharge. In small
rivers, on the other hand, the values of c vary little with the slope.
As regards the influence of roughness of the sides of the channel a
different law holds. For very small channels differences of roughness
have a great influence on the discharge, but for very large channels
different degrees of roughness have but little influence, and for
indefinitely large channels the influence of different degrees of
roughness must be assumed to vanish. The coefficients given by Darcy
and Bazin are different for each of the classes of channels of
different roughness, even when the dimensions of the channel are
infinite. But, as it is much more probable that the influence of the
nature of the sides diminishes indefinitely as the channel is larger,
this must be regarded as a defect in their formula.
Comparing their own measurements in torrential streams in Switzerland
with those of Darcy and Bazin, Ganguillet and Kutter found that the
four classes of coefficients proposed by Darcy and Bazin were
insufficient to cover all cases. Some of the Swiss streams gave
results which showed that the roughness of the bed was markedly
greater than in any of the channels tried by the French engineers. It
was necessary therefore in adopting the plan of arranging the
different channels in classes of approximately similar roughness to
increase the number of classes. Especially an additional class was
required for channels obstructed by detritus.